Electric potential and electric potential energy

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Examples
Lessons
1. Electric potential energy and electric potential between parallel plates (constant E)
There is a constant electric field of 11 500 N/C between a pair of oppositely-charged parallel plates spaced 0.0475 m apart.
1. Find the amount of work needed to move a proton from the surface of the negative plate to the surface of the positive plate, and how much electric potential energy the proton has gained after it is moved.
2. If a proton is held at the positive plate surface and released, find the speed of the proton when it reaches the surface of the negative plate.
3. Find the electric potential difference between the two plates. What potential energy would a $6.00 \times 10^{-7} C$ charge have at the positive plate surface?
2. Electric potential energy and electric potential of a point charge (non-constant E)
Find the electric potential at each point A and B, and the potential difference between A and B. If 0.060 J of work must be done on a charge to move it from A to B, find the magnitude and polarity of the charge.
Topic Notes
In this lesson, we will learn:
• The meaning of electric potential and electric potential energy
• How to understand electric potential energy problems by analogy to gravitation problems
Notes:

• Electric potential energy (Ep) is the energy stored in a charge due to its location in an electric field. For example, moving two like charges close together stores potential energy, since the charges repel each other. Separating opposite charges also stores potential energy, since the charges attract each other. It is similar idea to gravitational potential energy.
• It is useful to calculate changes in electric potential energy using the concept of electric potential difference ($\Delta V$). An electric potential ($V$) is the electric potential energy of a charge q, divided by q. $\Delta V$ is the difference in $V$ between two points.

Electric Potential Energy (Two Point Charges)

$E_p = k \frac{Q_1 Q_2}{r^2}$
$E_p:$ electric potential energy, in joules (J)
$k = 9.00 \times 10^9 N\centerdot m^2 / C^2$ (Coulomb's constant)
$Q_1, Q_2:$ charge on each body, in coulombs (C)
$r:$ distance between charges, in meters (m)

Electric Potential Energy (Charge in Constant Electric Field)

$E_p=qEd$
$E$: electric field, in newtons per coulomb (N/C)
$q$: charge that experiences the $E$, in coulombs (C)
$d$: distance from location chosen as $E_p=0$ J, meters (m)

Electric Potential

$V=\frac{E_p}{q}$
$V$: electric potential, in volts (V)
$E_p$: electric potential energy, in newtons (N)
$q$: charge that experiences the potential, in coulombs (C)

Electric Potential Difference

$\Delta V=V_f - V_i = \frac{W_{on\;q}}{q} = \frac{\Delta E_q}{q}$
$\Delta V$: electric potential difference, in volts (V)
$V_i,V_f$: electric potential at initial and final points, in volts (V)
$q$: charge that experiences the potential, in coulombs (C)
$W_{on\;q}$: work to move $q$ from the initial point to the final point, in joules (J)
$\Delta E_q$: change in potential energy of $q$ between the initial and final points, in joules (J)

Useful Constants

$k = 9.00 \times 10^9 N\centerdot m^2 / C^2$ (Coulomb's constant)
$e = 1.60 \times 10^{-19} C$ (Elementary charge. $q_{proton}=e, q_{electron}=-e$)
$m_{proton}=1.67 \times 10^{-27}$ kg
$m_{electron}=9.11 \times 10^{-31}$ kg